https://digital.csic.es/bitstream/10261/15682/1/blanes.pdf

The Magnus expansion and some of its applications

These notes were prepared for use by the participants in the Workshop on Algebra, Geometry and Topology held at the Australian National University, 22 January to 9 February, 1996. They have subsequently been updated for use by students in the subject 620-421 Combinatorial Group Theory at the University of Melbourne. Copyright 1996-2002 by C. F. Miller.

Combinatorial Group Theory

"Symmetry is a complexity-reducing concept [...]; seek it every-where." - Alan J. PerlisMany natural and man-made objects exhibit significant symmetries or contain repeated substructures. This paper presents a new algorithm that processes geometric models and efficiently discovers and extracts a compact representation of their Euclidean symmetries. These symmetries can be partial, approximate, or both. The method is based on matching simple local shape signatures in pairs and using these matches to accumulate evidence for symmetries in an appropriate transformation space. A clustering stage extracts potential significant symmetries of the object, followed by a verification step. Based on a statistical sampling analysis, we provide theoretical guarantees on the success rate of our algorithm. The extracted symmetry graph representation captures important high-level information about the structure of a geometric model which in turn enables a large set of further processing operations, including shape compression, segmentation, consistent editing, symmetrization, indexing for retrieval, etc.

/pdf/partial-and-approximate-symmetry-detection-for-3d-geometry-4ftkho2cwu.pdf

Partial and approximate symmetry detection for 3D geometry

We use Quillen's theorem and algebraic geometry to investigate the modular transformation properties of some quantities of interest in string theory. In particular, we show that the spin structure dependence of the chiral Dirac determinant on a Riemann surface is given by Riemann's theta function. We use this result to investigate the modular invariance of multiloop heterotic string amplitudes.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1104115581

Theta functions, modular invariance, and strings

An abelian quotient of the mapping class groupI g

Combinatorial Group Theory.

Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations

/pdf/note-on-a-theorem-of-schreier-12widvrcfm.pdf

A. Karrass

Papers

Combinatorial Group Theory.

Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations

Note on a theorem of Schreier

Some Remarks on the Infinite Symmetric Groups.

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